OFFSET
1,2
COMMENTS
Suppose that p is a partition of n, let F(p) be its Ferrers matrix, as defined at A237981, and let mXm be the size of F(p). The numbers of 1's in each of the 2m-1 antidiagonals of F(p) form a partition of n. Any partition which is associated with a partition of n in this manner is introduced here as an antidiagonal partition of n. A000041(n) = sum of the numbers in row n; A000009(n) = number of terms in row n, since the antidiagonal partitions of n are the conjugates of the strict partitions of n.
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..1000
Clark Kimberling and Peter J. C. Moses, Ferrers Matrices and Related Partitions of Integers
EXAMPLE
The Mathematica ordering of the 6 antidiagonal partitions of 8 follows: 3221, 32111, 22211, 221111, 2111111, 11111111. Frequencies of these among the 22 partitions of 8 are given in reverse Mathematica ordering as follows: 11111111 occurs 2 times, 2111111 occurs 2 times, 221111 occurs 4 times, 22211 occurs 6 times, 32111 occurs 2 times, and 3221 occurs 6 times, so that row 8 of the array is 2 2 4 6 2 6.
...
First 12 rows:
1;
2;
2, 1;
2, 3;
2, 2, 3;
2, 2, 6, 1;
2, 2, 4, 3, 4;
2, 2, 4, 6, 2, 6;
2, 2, 4, 4, 2, 3, 9, 4;
2, 2, 4, 4, 2, 6, 6, 3, 12, 1;
2, 2, 4, 4, 2, 4, 6, 3, 6, 6, 12, 5;
2, 2, 4, 4, 2, 4, 6, 6, 4, 6, 3, 18, 2, 4, 10;
MATHEMATICA
z = 20; ferrersMatrix[list_] := PadRight[Map[Table[1, {#}] &, #], {#, #} &[Max[#, Length[#]]]] &[list]; antiDiagPartNE[list_] := Module[{m = ferrersMatrix[list]}, Map[Diagonal[Reverse[m], #] &, Range[-#, #] &[Length[m] - 1]]]; a1[n_] := Last[Transpose[Tally[Map[DeleteCases[Reverse[Sort[Map[Count[#, 1] &, antiDiagPartNE[#]]]], 0] &, IntegerPartitions[n]]]]];
t = Table[a1[n], {n, 1, z}]; TableForm[Table[a1[n], {n, 1, z}]] (* A238325, array *)
u = Flatten[t] (* A238325, sequence *)
(* Peter J. C. Moses, 18 February 2014 *)
CROSSREFS
KEYWORD
nonn,tabf,easy
AUTHOR
Clark Kimberling and Peter J. C. Moses, Feb 24 2014
EXTENSIONS
Example corrected by Peter J. Taylor, Apr 10 2022
STATUS
approved